\section{Results}
\label{sec:result}

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/Ncoll.pdf}
  \caption{The differencial distribution of number of collision in different $P_{T}$ region. The distributions including different initial (drawn in dashed lines) or final (drawn in solid lines) $P_{T}$ shreshold. The black lines mean low $P_{T}$ region, where $P_{T}<0.5GeV$ and red lines mean high $P_{T}$ region, where $P_{T}>3GeV$, for both initial and final $P_{T}$ cuts.}
  \label{fig:ncoll}
  \end{center}
\end{figure}
The analysis of number of collision


Since we have got the "collision chain", we can get the number of collisions of every recorded parton, as well as the initial and final information of the parton. It's also a very important imformation of a parton.  
In Figure \ref{fig:ncoll}, we show the number of collision distribution. The dot lines 
mean distribution of intial partons and the active lines mean distribution of  the final partons. Partons with both high and
 low transverse momentum are showed seperately using red and black lines. The distribution of parton with low transverse momentum( $P_{T}<0.5GeV$ ) are similar. We know that the cross-section is indepandent to parton transverse momentum.  
The difference between partons with higher transverse momentum and lower transverse momentum is the time when the parton was generated.We can see it from Figure \ref{fig:pttime}, this is a 2D histogram of parton initial transverse momentum and the time when it was generated. It reveals that the parton with lower transverse momentum appears earlier so that it may have more chance to collide with other partons.
In fact the parton formation time from string-melting in AMPT was set as $tf=E_H/m_{T,H}^2$ where H represents the parent hadron of the parton. That also coincide with our conclusion. There will be more discussion about relationship between parton tranverse momentum and number of collision later.
\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/pt_time.pdf}
  \caption{This plot shows the relationship between parton formation time and parton initial $P_{T}$.}
  \label{fig:pttime}
  \end{center}
\end{figure}



The analysis of parton transverse momentum 


We can see it clearly in Figure \ref{fig:pt2d}.
In Figure \ref{fig:pt2d}, we show comparison of intial and final transverse momentum of partons. Most partons are with transverse momentum less than 1GeV. 
Parton with higher intial transverse momentum could collide more times because they have more energy to scatter with other partons. 
These partons are keeping lossing energy with the collisions which we can see from Fig.\ref{fig:pt2d}. The partons with high final transverse momentum are more complex. 
From Fig.\ref{fig:pt2d} we can see there are two kinds of these partons. The first kind is from low initial $P_{T}$ which gain energy from the collisions. 
And the second kind comes from the high $P_{T}$ partons which loss energy. 
\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/pt2d.pdf}
  \caption{2D plots for initial and final parton $P_{T}$. The x-axis is the partons' initial $P_{T}$ and y-axis is partons' final $P_{T}$.}
  \label{fig:pt2d}
  \end{center}
\end{figure}

In order to check how does the partons with low transverse momentum gain energy in collisions, we plot Figure. \ref{fig:pt2d_special}. This plot has two differences with last one. Firstly the partons in this plot is with $P_{T}^{Initial}<1GeV$ and  $P_{T}^{Final}>3GeV$. Secondly, the number of collision in this plot is not the number which means how many times does a proton collide with others. The number of collision means how many collision does the parton have in this time. For example, if a parton has 3 collisions, it will be filled 4 times in this plots at bin is 0,1,2 and 3. 
We will use this definition many times later, so we define it $N_{coll}^{*}$ so as to distinguish with $N_{coll}$. Then let's come back to Figure. \ref{fig:pt2d_special}. The conclusion we can get from it is that a lot of parton get huge energy via only 1 collision! 
This can be also checked by Figure. \ref{fig:maxdpt}, it shows the max momentum the parton gains in all collisions. There is a peak ( I thought it should be sharper) at about 3GeV which shows alot of partons' energy change from 1GeV to 3GeV by just 1 collision.
\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/checktracing_ptn_sct.pdf}
  \caption{2D plot of parton $P_{T}$ vesus $N^{*}_{coll}$ in phase space: $P_T^{Initial}<1GeV$ and $P_T^{final}>3GeV$.}
  \label{fig:pt2d_special}
  \end{center}
\end{figure}

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/dpt_max.pdf}
  \caption{The distribution of max $P_{T}$ change of a parton in its collisions. For a certain parton, the $P_{T}$ of it changes in every collision, where the max $\Delta P_{T}$ here means the max value of the $P_{T}$ change of the parton.}
  \label{fig:maxdpt}
  \end{center}
\end{figure}



Relationship between transverse energy and number of collision

In order to research on how the energy change in prosess of collisions.  
Figure \ref{fig:dpt_vs_N} is difference of final and initial transverse momentum as a function of number of collision. We compare their relationship with different $P_{T}$ shresholds to see the energy change of partons with different energy.
The black line shows the energy gaining with initial $P_{T}<0.2GeV$, which is smaller than the averge transverse energy about 0.3GeV. The colorful lines show high $P_{T}$ intial partons loss energy and we can see the higher intial $P_{T}$, the more energy they loss. That's as what we expected.  
We can go a little further. Next step we consider a new parameter we call it "$R_{P_{T}}$". The define the ratio as $(P_{T}^{final}-P_{T}^{initial})/(P_{T}^{initial}-<P_{T}>)$, 
where $<P_{T}>=0.308GeV$ is the average transverse momentum. This quantity is the fractional energy loss or gain, which is better thing is this parameter is scaled by $(P_{T}^{initial}-<P_{T}>)$ which is negative for low energy partons but positive for high energy partons.
So that this parameter will always be negative. We can see Figure. \ref{fig:rpt_vs_N}. To better evaluate the numbers in this plot, we have to compare with the ratio value with -1. Where "-1" means the parton's final transverse momentum is equal to the mean transverse momentum of all the partons. We expect that curves with different intial energy to be flat with value "-1" however the fact does not go like that. The black line which mean partons with initial $P_{T}<0.2GeV$ comese to be flat after about 3 times collisions. And the value of that ratio is larger than "-1" means the final transverse momentum of these partons is smaller than the mean transverse momentum. The other colorful lines reveal different things. The blue and green lines which represent partons with intial $P_{T}>3GeV$ and $P_{T}>2GeV$ seperately stabalized at value larger than "-1" means the final transverse momumtem is larger than average. So that makes sense that not all the partons with different initial energy come to uniform after collisions. The partons with higher energy come up with higher final energy in statistic.
%What's interesting is the low $P_{T}$ partons. Combine the Fig.\ref{fig:pt2d} again, we can see in $P_{T}<0.5GeV$ region, not all partons gain energy by collsion. 
%This dues to the everage final $P_{T}$ is less than 0.5GeV. The trend is always partons gain less or loss more energy as the number of collision is larger. The energy of parton is approaching balance after some collisions. ( DO WE NEED MORE DETAILED ANALYSIS, FOR EXAMPLE, ENERGY GAIN IN EVERY STEP?)





\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/dpt_vs_N.pdf}
  \caption{$P_{T}^{Final}-P_{T}^{Initial}$ as a function of number of collision in different initial $P_{T}$ region.}
  \label{fig:dpt_vs_N}
  \end{center}
\end{figure}

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/n_vs_dpt.pdf}
  \caption{Average number of collisions vesus $P_{T}^{Final}-P_{T}^{Initial}$ in different initial $P_{T}$ region.}
  \label{fig:n_vs_dpt}
  \end{center}
\end{figure}


\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/rpt_vs_N.pdf}
  \caption{$(P_{T}^{final}-P_{T}^{initial})/(P_{T}^{initial}-<P_{T}>)$ as a function of number of collision in different initial $P_{T}$ region.}
  \label{fig:rpt_vs_N}
  \end{center}
\end{figure}


%\begin{figure}[!htbp]
%  \begin{center}
%  \includegraphics[width=0.9\columnwidth]{figures/phi2d.pdf}
%  \caption{2D plots for initial and final parton $\phi$.}
%  \label{fig:phi2d}
%  \end{center}
%\end{figure}

%The Figure \ref{fig:phi2d} show comparison of final and initial azimuthal angle. 
%There are two kinds of partons. The first kind is partons which do not change the direction when collide. 
%It is shown in the middle region of Fig \ref{fig:phi2d}. The second kind is parton with collision that change the direction of parton. 
%We can see them at the left-upper and right-bottom of Fig \ref{fig:phi2d}.

%\begin{figure}[!htbp]
%  \begin{center}
%  \subfigure[$N_{coll}=1$]{
%    \includegraphics[width=0.4\columnwidth]{figures/phi2d1.pdf}
%    \label{fig:phi2d1}
%  }
%  \subfigure[$N_{coll}=3$]{
%    \includegraphics[width=0.4\columnwidth]{figures/phi2d3.pdf}
%    \label{fig:phi2d3}
%  }
%  \end{center}
%  \caption{2D plots for initial and final parton $\phi$ when $N_{coll}=1$ and $N_{coll}=3$}
%i%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Analysis of phi distribution

Before we step into study about v2 distribution, we can generally have a glance at information about phi, because v2 is strongly related to phi.
In Figure \ref{fig:init_phi_N}, we want to research on the relationship of azimuthal angle and number of collisions.
We choose the phase space $1GeV<P_{T}<2GeV$ because the energy of partons in this phase space is neither too high or too low. The partons with few collisions distribute along the short axis of the elliptic.
It is due to there is less overlap of two nucleis in this region. So the number of collisions is less than long axis of the elliptic. For the same reason, we can see v2 is getting less as the number of collission being larger. 
\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/init_phi_N.pdf}
  \caption{Initial $\phi$ distribution with different collision number in $1GeV<$initial $P_{T}<2GeV$ region}
  \label{fig:init_phi_N}
  \end{center}
\end{figure}

Analysis of v2

Since v2 is an important parameter reveals the construction of the flow, we will discuss the it related with number of collision( both $N_{coll}$ and $N_{coll}^{*}$) and transverse momentum respectively.
Firstly we look at the relationship of v2 and number of collision. Similar with the analysis we did before we study the two definitions of "number of collision" so that it shows more useful information. 

Before all the discussion, we have to illustrate how we get the v2 value. In fact we have 3 methods to get the v2 value, they should be equivalence to each other mathematically. But subtle difference due to the restriction of the methods.
The first way is that we fit the $cos(\phi )$ distribution of partons with different number of collisions in different $P_{T}$ region. The fitting function we use is $p_{0}*[1+2p_{1}cos*(2\phi )]$, where $p_{1}$ is v2.
The second method is to calculate the mean value of phi which is the definition of v2 and the third method is similar to second one, that we use TProfile to do the same thing so that we can read both the mean value and the statistic uncertainty from it. 

\begin{table}[ht]
\small
\caption{v2 of initial parton(N=0) using different method}
\centering
\begin{tabular}{c c c c c}
\hline\hline
Method & (0,0.5GeV) & (0.5,1GeV) & (1GeV,2GeV) & (2GeV,10GeV)\\
\hline
Mean of phi & -1.4821e-6 & -1.8793e-4 & -7.3344e-4 & -3.9503e-4\\
fitting & $2.181e-7 \pm 1.647e-5$ & $-1.975e-4 \pm 3.32e-5$ & $-7.751e-4 \pm 8.78e-5$ & $-3.256e-4 \pm 3.435e-4$ \\
TProfile & $5.9e-6 \pm 1.605e-5$ & N/A &  $-7.728e-4 \pm 8.554e-5$ & $-2.264e-4 \pm 3.348e-4$\\
\hline
\end{tabular}
\end{table} 
In this table, we can compare with the v2 result with partons $N_{coll}=0$ in different $P_{T}$ shreshold. We can see the mean value is not beyong the uncertainty which means the three methods coincide with each other. In this analysis we will use the third method which is TProfile one for the reason that 1) we can not get the uncertainty from 2ed; 2) the fitting has more system uncertainty due to the width of the bin of phi histogram.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
  \begin{center}
  \subfigure[Initial pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_1.pdf}
    \label{fig:v2_vs_N}
  }
  \subfigure[final pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_2.pdf}
    \label{fig:v2_vs_N_1}
  }
  \subfigure[Initial pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_i.pdf}
    \label{fig:v2_vs_N}
  }
  \subfigure[final pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_f.pdf}
    \label{fig:v2_vs_N_1}
  }

  \subfigure[Initial pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_i_prof.pdf}
    \label{fig:v2_vs_N}
  }
  \subfigure[final pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_f_prof.pdf}
    \label{fig:v2_vs_N_1}
  }
%  \subfigure[Delta v2]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_2.pdf}
%    \label{fig:v2_vs_N_2}
%  }
  \caption{v2 of Initial and final partons as a function of number of collision in different $P_{T}$ region. The left column is with initial $P_{T}$ cuts and right column is with final $P_{T}$ cuts. The three rows are for three different method to get this plot, which are 1) fitting with phi distribution; 2) calculate the mean value of $cos(2*\phi)$; 3) filling TProfile of $cos(2*\phi)$. In each plot dashed lines are partons initial v2 and solid lines are partons final v2. Different $P_{T}$ are shown with different colors. }
  \end{center}
\end{figure}
  
\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/v2_vs_n.pdf}
  \end{center}
  \caption{This plot shows parton's v2 vesus $N_{coll}^{*}$. We make this plot in different $P_{T}$ regions (dashed lines for initial $P_{T}$ and solid lines for final $P_{T}$  with different colors.}
\end{figure}


The initial v2 distribution coincides with our expectation from Fig.\ref{fig:init_phi_N}. 
Different $P_{T}$ region also affact relationship of v2 and number of collisions. Since the higher $P_{T}$ the more collisions happen, the v2 is larger as $P_{T}$ when there is no collision,
which means more partons along the short axis have collision. And the v2 decreases with the number of collisions because the number of partons is large in short axis. However, 
since higher $P_{T}$ partons always collide more times, the v2 will decease more slowly than low $P_{T}$ partons. After some collisions, the partons are going along the long axis due the flow theory. So the v2 goes up after a turning point.


We can also draw similar conclusion from Fig.\ref{fig:v2_vs_pt}. This figure shows the relationship of v2 and parton $P_{T}$ with different number of collisions. The parton initial v2 compares with initial $P_{T}$( shows in dot lines) and final v2 compares with final $P_{T}$( shows in active lines).
We can see that in region $P_{T}<0.5GeV$, $v2_{N=1}<v2_{N=2}<v2_{N>2}$. And in regiond $P_{T}>1GeV$, $v2_{N=1}>v2_{N=2}>v2_{N>2}$, which coincides with Fig.\ref{fig:v2_vs_N}.
Fig.\ref{fig:v2_vs_N_1} is similar with Fig.\ref{fig:v2_vs_N}. The difference is the the dot lines are  initial v2 with different final $P_{T}$ region, which is initial $P_{T}$ region in Fig.\ref{fig:v2_vs_N}.
The final v2 distribution can be studied with the active lines in \ref{fig:v2_vs_N_1} and \ref{fig:v2_vs_pt_1}.
In \ref{fig:v2_vs_pt_1} the final v2 increases firstly because the flow will expand along the short axis and the higher the energy parton has the strong the expansion be. And the final v2 drops due to the elliptic flow will stay equilibrium after collide many times.
What's interesting is that the slope of increase and decrease is not related to final parton $P_{T}$. We can see it from Fig.\ref{fig:v2_vs_pt_3}, which is $\Delta v2=v2_{final}-v2_{initial}$ vsfinal $P_{T}$.
It shows the slope of $\Delta v2$ is same exept for the $P_{T}$ where v2 gets maximum. It dues to high energy parton collides more times.  
And Fig.\ref{fig:v2_vs_N_2} shows the $\Delta v2=v2_{final}-v2_{initial}$ vs number of collisionsin different final pT region. We can see that the $\Delta v2$ does not changed after 2 or 3 collision. 
Which means after 2 or 3 collisions, the flow has build up. 

 
\begin{figure}[!htbp]
  \begin{center}
  \subfigure[Initial pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_1.pdf}
    \label{fig:v2_vs_pt}
  }
  \subfigure[Final pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_2.pdf}
    \label{fig:v2_vs_pt_1}
  }
  \subfigure[Initial pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_i.pdf}
    \label{fig:v2_vs_pt}
  }
  \subfigure[Final pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_f.pdf}
    \label{fig:v2_vs_pt_1}
  }
  \subfigure[Initial pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_i_prof.pdf}
    \label{fig:v2_vs_pt}
  }
  \subfigure[Final pt for v2]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_f_prof.pdf}
    \label{fig:v2_vs_pt_1}
  }
%  \subfigure[Delta v2]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_2.pdf}
%    \label{fig:v2_vs_pt_2}
%  }
  \caption{V2 of partons' initial and final states as a function of parton $P_{T}$ with different $N_{coll}$ (labled in different colors).The x-axises of left column is partons' initial $P_{T}$ and the x-axises of right column is partons' final $P_{T}$. The partons' initial v2 are drawn in dashed lines and final v2 are drawn in solid lines.}
  \end{center}
\end{figure}
  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{figure}[!htbp]
\begin{figure}[!htbp]
  \begin{center}
  \subfigure[phi distribution]{
    \includegraphics[width=0.45\columnwidth]{figures/test4.pdf}
    \label{fig:v2_vs_N_i}
  }
  \subfigure[Ncoll = 0, 1<pt<2]{
    \includegraphics[width=0.45\columnwidth]{figures/test2.pdf}
    \label{fig:v2_vs_N_i}
  }
  \subfigure[Ncoll = 0, pt>2]{
    \includegraphics[width=0.45\columnwidth]{figures/test3.pdf}
    \label{fig:v2_vs_N_i}
  }
  \subfigure[at the start]{
    \includegraphics[width=0.45\columnwidth]{figures/test1.pdf}
    \label{fig:v2_vs_N_f}
  }
  %\caption{v2 of Initial and final partons as a function of number of collision in different $P_{T}$ region.}
  \caption{test}
  \end{center}
\end{figure}

%\begin{figure}[!htbp]
%  \begin{center}
%  \subfigure[Initial pt for v2]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_i.pdf}
%    \label{fig:v2_vs_pt}
%  }
%  \subfigure[Final pt for v2]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_f.pdf}
%    \label{fig:v2_vs_pt_1}
%  }
%  \caption{v2 of Initial and final partons as a function of parton $P_{T}$ with different $N_{coll}$.}
%  \end{center}
%\end{figure}


